Abstract:

Rigorous ray tracing originates from the trajectory representation of quantum mechanics while the (Leontovich--Fock) parabolic equation was originally applied to electromagnetic propagation. Rigorous ray tracing (and quantum trajectory representation) have had theoretical success while the parabolic approximation has been a computational success. The parabolic approximation inspires a similar equivalent approximation to the generalized Hamilton--Jacobi equation for rigorous ray tracing. For separable coordinates, the resulting approximations render the same approximate wavefunction. In return, the rigorous ray leads to a parabolic approximation incorporating reflection. The trajectory representation has rendered a synthesized single wavefunction that includes an incident wave of coefficient (alpha) and a reflected wave of coefficient (beta). This synthesized wavefunction can generate a new alternative parabolic equation that incorporates reflection, which for small reflection ((beta)<<(alpha)) becomes F[inf zz]+i2k[1-(2(beta)/(alpha))cos(2kx)]exp[i(2(beta)/(alpha))sin(2kx)]F[inf x]+((kappa)[sup 2]-k[sup 2])F=0.